train stopovers

In Germany, you can specify a stopover for several hours on a Super Saver Ticket (advance purchase nonrefundable) on national trains (ICE, IC, EC). I forget the time limit, but I believe it can be until the last train of the day.
If you continue on a regional train, you can take any regional train after the national train. Must be shown on the original ticket. The journey is completed by 10 am the next day.

Train-specific travel
If you have a super saver fare ticket, you can only travel on the service indicated on the ticket. The offers are available for ICE, IC/EC connections within Germany. You can use local rail services (e.g. RE, RB, IRE and S-Bahn or non-DB trains) to get to/from the long-distance station if you include them in your booking. You are not restricted to travelling on a specific train when using local rail services. However, your journey must be completed by 10 am the following day.

If you take only regional trains, you can stop off anywhere along the route, and continue the journey on any regional train.

If you are traveling in a given Land (roughly equivalent to a US state), you can buy a Laender ticket and travel on any regional train in that Land until 3 am the next day. You have to travel weekday after 9 am, or anytime weekends and holidays. In a big Land like Bavaria, that can be very useful. That also includes local transport like U-bahns, trams, buses.
https://www.bahn.com/en/offers/regional

In Switzerland, you can stop over anywhere along the route as long as you do not have a discounted train-specific ticket.

You don't specify which country you are speaking of.

In the UK there are some tickets which allow in one direction, some in both, and many not at all. It is important to be sure you have the correct ticket - there are many.

Regional trains in Italy have some flexibility, generally good for 4 hours from the time you validate the ticket in the platform stamping machine. Since the trains are subsidized by the provincial governments, these rules can vary depending on where you are in Italy. So short stopovers are possible.

TER trains in France are good for 24 hours from the time you validate your ticket in the platform stamping machine. At least that was the case 10 years ago, but I can't find the current conditions of carriage on line.

Belgian trains are generally good for a stopovers, except for high speed trains like the Thalys. Also, there is a surcharge for stopping at Brussels airport.

Netherlands trains run like a suburban network where you need to scan your ticket to enter and exit the station platform, so stop overs are problematic, but then the cost of buying individual tickets is not a problem.

.....passes allowing unlimited travel...... Not sure what you mean by that or your understanding. Generally speaking -- in broad terms because every train system is different -- a train pass does not convey the right jump on any train at any time. Often the train will require a seat reservation -- at an additional cost -- along with the train pass. So if you got off the train you would need a new seat reservation to get back on later. However, some trains system run a mix of reserved and non-reserved seats so you could do that on those trains. You might have to stand also if you cannot find an open seat. So without knowing which countries you are traveling through, the response can only be --- It depends.

And do the math. A rail pass may or may not be the most economic choice. These days the advance discount prices for tickets may under cut a pass.

Just out of curiosity, I queried the Bahn schedule and fare webpage (reiseankunft.bahn.de/bin/query.exe/en) for a connection leaving Munich after 14:00 going to Frankfurt Flughafen with a 17 hour stopover in Würzburg.

The first connection the Bahn offered was leaving Munich at 14:45, arriving FRA at 14:45 the next day. It only offered one fare, a flexipreis (standard) ticket for 98,50 EUR. When I selected "earlier", it offered me a connection leaving Munich at 12:45, arriving at Würzburg at 14:52. The stopover in Würzburg was 17 hr, 32 min, leaving Würzburg at 8:24 the next morning, arriving at FRA at 9:56. For that connection, it offered both a flexipreis of 98,50 EUR and a Super Sparpreis fare of 23,90 EUR (also a Sparpreis fare of 29,90).

If you take only regional trains, you can stop off anywhere along the
route, and continue the journey on any regional train.

I think the same 100km rule applies to regional tickets (less than 100km, finish by end of day; more than 100km, by end of 2nd day). On a regional pass (eg, Bayern-Ticket), you must finish your travel by 3 AM the next morning.

I've tried the same thing multiple times in the past, putting in a stopover giving me time for an overnight stay, with similar results. My advice, don't ask us here, try it yourself with an inquiry on the Bahn website.

BTW, if you tried that connection with a railpass, it would use two days of your pass.

I've only had this once and it was a fairly long time ago (~15 years). We were traveling Zagreb to Salzburg and were stopping in Ljubljana for a couple of days. I thought I needed to buy two separate tickets. The tourist office in Zagreb guided me to a travel agency who sold me a ticket that was Zagreb to Salzburg and indicated I could stop anywhere along the way (don't remember what the time limit was). The caveat was that I needed to pay for two separate reservations for the 2 legs I was intending to ride. Overall, it saved me some money.

I don't know if these types of tickets still exist. I've searched on the internet years ago but never was able to confirm if these are still around. Will probably need to go into a major railway booking office in Europe and ask.

The type of international ticket with stopovers allowed along a prescribed route still exists when purchased at a European train station, but most likely to work the farther east you go, to areas where trains are generally slower and tickets are not required to be sold with reservations built in. These international tickets are now limited to travel within 3 days, down from 14 days some years ago.

Germany, Austria, and Switzerland also don’t tend to require train reservations, but selecting a specific train in advance can afford significant discounts over the full fare.

Within one country, the time to complete a journey on unreserved trains is normally less than 24 hours, and sometime a lot less, depending on distance and country.

One wrinkle not mentioned, is the growing use of e-tickets on your phone. In Italy at least. it can mean that even a Regional ticket can wind up being specific to the train and time, as opposed to an open regional ticket for that route. To use for a different train may require you to go in and modify your journey to match the train you want to get on.

I suppose this is my cue to wax romantically about the Kilo Ticket Italy had a couple decades ago, you paid a flat fee for 1000 kilometers of travel, listed the day, start station, and destination, the conductor deducted the distance from your total, it could even be used for more than one person. Easy and economical.

it can mean that even a Regional ticket can wind up being specific to the train and time, as opposed to an open regional ticket for that route.

In Italy the regional trains e-tickets are less flexible than the paper version, but they are not specific to only one train. Since you can get online a train ticket up to 15 minutes before the departure time and Regional trains have no reserved seats, the difference has no bearing.

To use for a different train may require you to go in and modify your journey to match the train you want to get on.

Not in Italy. You'd just get on a Regional train making the same route and departing within the next four hours.

BTW, I use this link, https://reiseauskunft.bahn.de/bin/query.exe/en, for Bahn schedules and ticket prices. There are other, perhaps easier to remember, routes to get to this webpage, but this link gets you there directly, and in English. (Although I can read German, I usually go to the English webpage when giving advice here, so I can see what you English-only users will see.)

Or you could ask a country- and time-specific question to that Man in Seat 61 on his website, he's pretty responsive

Fly in the ointment here...

What do you do with your luggage (carry-on and personal item, of course 😉) while you are stopping over? Not every train station has places to check it.

I have to think all midsize and large size stations have facilities for left luggage. Otherwise, perhaps look for a nearby hotel and pay them to babysit your bags?